DUBLIN CITY UNIVERSITY

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1 DUBLIN CITY UNIVERSITY SAMPLE EXAMINATIONS 2017/2018 MODULE: QUALIFICATIONS: Simulation for Finance MS455 B.Sc. Actuarial Mathematics ACM B.Sc. Financial Mathematics FIM YEAR OF STUDY: 4 EXAMINERS: Mr Denis Patterson (ext. 7710), External Examiner TIME ALLOWED: INSTRUCTIONS: REQUIREMENTS: 2 hours Answer ALL 4 questions Approved calculators and Log tables PLEASE DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO The use of programmable or text storing calculators is expressly forbidden. Please note that where a candidate answers more than the required number of questions, the examiner will mark all questions attempted and then select the highest scoring ones. MS455, Sample 2017/2018 Page 1 of 5

2 QUESTION 1 (a.) A Linear Congruential Generator is a sequence {u i } i 0 generated by an iteration of the form x i+1 = (ax i + c) mod m, i {0, 1,...}, u i+1 = x i+1 m (0, 1), i {0, 1,...}, a, c, m Z+. State precisely necessary and sufficient conditions for a Linear Congruential Generator to have a full period. [6 marks] (b.) S = {x 1,..., x n } is a sample of uniform random numbers on [0, 1]. Explain how you would perform a Kolmogorov Smirnov test for uniformity on the sample S. [7 marks] (c.) A sample of 100 pseudorandom numbers which are (supposedly) distributed uniformly on [0, 1] was generated. The interval [0, 1] was divided into 10 equally spaced subintervals and the number of pseudorandom numbers falling into each subinterval was recorded (see the table below). Table 1: The first (resp. third) row indicates the subinterval and the second (resp. fourth) row indicates the number of sample values in that subinterval. [0, 0.1] [0.1, 0.2] [0.2, 0.3] [0.3, 0.4] [0.4, 0.5] [0.5, 0.6] [0.6, 0.7] [0.7, 0.8] [0.8, 0.9] [0.9, 1] Perform a Chi Square test for uniformity on the sample considered above at the 95% confidence level. [12 marks] HINT: Clearly state the null and alternative hypotheses, and whether or not the null hypothesis was accepted or rejected. MS455, Sample 2017/2018 Page 2 of 5

3 QUESTION 2 (a.) Outline the algorithm for the Inverse Transform Method and prove that it produces random variables with the correct distribution. (b.) Consider a one sided Normal distribution with density 2 /2σ f(x) = 2 πσ 2 e x2, x 0, σ > 0, and assume that you can sample from an exponential distribution with any parameter λ > 0. Which exponential distribution yields the most efficient Acceptance rejection scheme for the Normal distribution given above? [15 marks] HINT: The p.d.f. of an exponential random variable with parameter λ > 0 is given by g(x) = λe λx, x 0. MS455, Sample 2017/2018 Page 3 of 5

4 QUESTION 3 Consider the definite integral I = 0 x 2 e x dx. (a.) Explain how you would estimate the integral above by interpreting it as the expectation of a Weibull distributed random variable. You should clearly state the Monte Carlo estimator you would use for I. HINT: The p.d.f. of a Weibull random variable with parameters k > 0 and λ > 0 is given by { k ( x ) k 1 f(x) = λ λ e (x/λ) k, x 0, 0, x < x 0. (b.) Calculate the mean and variance of the Monte Carlo estimator of I and use the Central Limit Theorem to give an asymptotic 95% confidence interval for the Monte Carlo estimate of I. [15 marks] MS455, Sample 2017/2018 Page 4 of 5

5 QUESTION 4 Consider an autonomous scalar stochastic differential equation of the form dx t = a(x t ) dt + b(x t ) db t, t 0. (SDE) (a.) State precisely sufficient conditions on the functions a and b for (SDE) to have a unique solution. [7 marks] (b.) Derive the Euler Maruyama discretisation scheme for (SDE) and outline its most important performance characteristics. You should define any terminology you introduce in your explanation. (c.) Let {B t, F t : t 0} be a standard Brownian motion and suppose the process Y obeys the stochastic differential equation dy t = θdb t 1 2 θ2 dt, t 0; Y 0 = 0, for some θ R/{0}. Use Itô s Lemma to derive the stochastic differential equation satisfied by X t = e Yt for t 0 and hence show that E[X t ] = 1 for each t 0. [8 marks] MS455, Sample 2017/2018 Page 5 of 5